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<h1 class="title toc-ignore">Simulating Design Matrix <span class="math inline">\(X\)</span>: Correlation patterns</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2018-02-05</em></h4>
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<p><strong>Last updated:</strong> 2018-05-15</p>
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<hr />
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>For a fixed design matrix <span class="math inline">\(X\)</span>, <span class="math inline">\(\hat\beta \sim N(\beta, \sigma_e^2(X^TX)^{-1})\)</span>, and the empirical distribution of <span class="math inline">\(\hat z_j = \hat\beta_j / \hat{\text{SE}}(\hat\beta_j)\)</span> under the null will depend on the average correlation in <span class="math inline">\((X^TX)^{-1}\)</span>.</p>
<p>Actually the most important quantity to determine the shape of the empirical distribution of <span class="math inline">\(\hat z_j\)</span> is the square root of the mean squared correlation among <span class="math inline">\(\hat\beta_j\)</span>’s, that is, <span class="math inline">\(\sqrt{\overline{\rho_{\hat\beta_i, \hat\beta_j}^2}}\)</span>.</p>
<p>Now we are taking a look at this quantity for some commonly used design matrix in linear regression simulations.</p>
<p>In all three settings, each row of <span class="math inline">\(X_{n \times p}\)</span> is independently drawn from a <span class="math inline">\(N(0, \Sigma)\)</span> distribution, where the diagonal elements of <span class="math inline">\(\Sigma\)</span> are all one. Then the columns of <span class="math inline">\(X\)</span> are normalized such that <span class="math inline">\(\|X_j\|_2^2 = 1\)</span>.</p>
</div>
<div id="independent-and-normalized-columns" class="section level2">
<h2>Independent and normalized columns</h2>
<ul>
<li><span class="math inline">\(\Sigma = I\)</span>.</li>
</ul>
<pre><code>[1] 0.02234663</code></pre>
<p><img src="figure/design_matrix_2.rmd/unnamed-chunk-2-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="toeplitz-column-correlation" class="section level2">
<h2>Toeplitz column correlation</h2>
<ul>
<li><span class="math inline">\(\Sigma_{ij} = \rho^{|i - j|}\)</span></li>
</ul>
<div id="textselefthatbeta_jright" class="section level3">
<h3><span class="math inline">\(\text{SE}\left(\hat\beta_j\right)\)</span></h3>
<p><img src="figure/design_matrix_2.rmd/unnamed-chunk-4-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-4-1.png:</em></summary>
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0d440454746b4d41353ce970834d535ccea9009e/docs/figure/design_matrix_2.rmd/unnamed-chunk-4-1.png" target="_blank">0d44045</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-06
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/design_matrix_2.rmd/unnamed-chunk-4-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-4-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
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<a href="https://github.com/LSun/truncash/blob/0d440454746b4d41353ce970834d535ccea9009e/docs/figure/design_matrix_2.rmd/unnamed-chunk-4-2.png" target="_blank">0d44045</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-06
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="average-orrelation-among-x_j-and-hatbeta_j" class="section level3">
<h3>Average orrelation among <span class="math inline">\(X_j\)</span> and <span class="math inline">\(\hat\beta_j\)</span></h3>
<p><img src="figure/design_matrix_2.rmd/unnamed-chunk-5-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-5-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
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<th style="text-align:left;">
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<th style="text-align:left;">
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</thead>
<tbody>
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0e2142f6f35a5f29470346aa1790d31f57edfc44/docs/figure/design_matrix_2.rmd/unnamed-chunk-5-1.png" target="_blank">0e2142f</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-08
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0d440454746b4d41353ce970834d535ccea9009e/docs/figure/design_matrix_2.rmd/unnamed-chunk-5-1.png" target="_blank">0d44045</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-06
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/design_matrix_2.rmd/unnamed-chunk-5-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-5-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
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<th style="text-align:left;">
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</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0e2142f6f35a5f29470346aa1790d31f57edfc44/docs/figure/design_matrix_2.rmd/unnamed-chunk-5-2.png" target="_blank">0e2142f</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-08
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0d440454746b4d41353ce970834d535ccea9009e/docs/figure/design_matrix_2.rmd/unnamed-chunk-5-2.png" target="_blank">0d44045</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-06
</td>
</tr>
</tbody>
</table>
</details>
</div>
</div>
<div id="factor-model-column-correlation" class="section level2">
<h2>Factor model column correlation</h2>
<ul>
<li><span class="math inline">\(\Sigma_X = \texttt{cov2cor}(B_{p \times d}B_{d\times p}^T + I)\)</span></li>
</ul>
<div id="textselefthatbeta_jright-1" class="section level3">
<h3><span class="math inline">\(\text{SE}\left(\hat\beta_j\right)\)</span></h3>
<p><img src="figure/design_matrix_2.rmd/unnamed-chunk-7-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-7-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
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<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/5cb719f99a8959aecb622965580fc17a6c1df27d/docs/figure/design_matrix_2.rmd/unnamed-chunk-7-1.png" target="_blank">5cb719f</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-06
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0d440454746b4d41353ce970834d535ccea9009e/docs/figure/design_matrix_2.rmd/unnamed-chunk-7-1.png" target="_blank">0d44045</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-06
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/design_matrix_2.rmd/unnamed-chunk-7-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-7-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
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</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/5cb719f99a8959aecb622965580fc17a6c1df27d/docs/figure/design_matrix_2.rmd/unnamed-chunk-7-2.png" target="_blank">5cb719f</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-06
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0d440454746b4d41353ce970834d535ccea9009e/docs/figure/design_matrix_2.rmd/unnamed-chunk-7-2.png" target="_blank">0d44045</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-06
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="average-correlation-among-x_j-and-hatbeta_j." class="section level3">
<h3>Average correlation among <span class="math inline">\(X_j\)</span> and <span class="math inline">\(\hat\beta_j\)</span>.</h3>
<p><img src="figure/design_matrix_2.rmd/unnamed-chunk-8-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-8-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
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</th>
<th style="text-align:left;">
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</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0e2142f6f35a5f29470346aa1790d31f57edfc44/docs/figure/design_matrix_2.rmd/unnamed-chunk-8-1.png" target="_blank">0e2142f</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-08
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0d440454746b4d41353ce970834d535ccea9009e/docs/figure/design_matrix_2.rmd/unnamed-chunk-8-1.png" target="_blank">0d44045</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-06
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/design_matrix_2.rmd/unnamed-chunk-8-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-8-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
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</th>
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</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0e2142f6f35a5f29470346aa1790d31f57edfc44/docs/figure/design_matrix_2.rmd/unnamed-chunk-8-2.png" target="_blank">0e2142f</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-08
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0d440454746b4d41353ce970834d535ccea9009e/docs/figure/design_matrix_2.rmd/unnamed-chunk-8-2.png" target="_blank">0d44045</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-06
</td>
</tr>
</tbody>
</table>
</details>
</div>
</div>
<div id="factor-model-hatbeta-correlation" class="section level2">
<h2>Factor model <span class="math inline">\(\hat\beta\)</span> correlation</h2>
<ul>
<li><span class="math inline">\(Cor(\hat\beta) = \texttt{cov2cor}(B_{p \times d}B_{d\times p}^T + I)\)</span></li>
</ul>
<div id="textselefthatbeta_jright-2" class="section level3">
<h3><span class="math inline">\(\text{SE}\left(\hat\beta_j\right)\)</span></h3>
<p><img src="figure/design_matrix_2.rmd/unnamed-chunk-10-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-10-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
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</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0e2142f6f35a5f29470346aa1790d31f57edfc44/docs/figure/design_matrix_2.rmd/unnamed-chunk-10-1.png" target="_blank">0e2142f</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-08
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/design_matrix_2.rmd/unnamed-chunk-10-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-10-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0e2142f6f35a5f29470346aa1790d31f57edfc44/docs/figure/design_matrix_2.rmd/unnamed-chunk-10-2.png" target="_blank">0e2142f</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-08
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="average-correlation-among-x_j-and-hatbeta_j.-1" class="section level3">
<h3>Average correlation among <span class="math inline">\(X_j\)</span> and <span class="math inline">\(\hat\beta_j\)</span>.</h3>
<p><img src="figure/design_matrix_2.rmd/unnamed-chunk-11-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-11-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0e2142f6f35a5f29470346aa1790d31f57edfc44/docs/figure/design_matrix_2.rmd/unnamed-chunk-11-1.png" target="_blank">0e2142f</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-08
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/design_matrix_2.rmd/unnamed-chunk-11-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-11-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0e2142f6f35a5f29470346aa1790d31f57edfc44/docs/figure/design_matrix_2.rmd/unnamed-chunk-11-2.png" target="_blank">0e2142f</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-08
</td>
</tr>
</tbody>
</table>
</details>
</div>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.4
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
loaded via a namespace (and not attached):
[1] workflowr_1.0.1 Rcpp_0.12.16 digest_0.6.15
[4] rprojroot_1.3-2 R.methodsS3_1.7.1 backports_1.1.2
[7] git2r_0.21.0 magrittr_1.5 evaluate_0.10.1
[10] stringi_1.1.6 whisker_0.3-2 R.oo_1.21.0
[13] R.utils_2.6.0 rmarkdown_1.9 tools_3.4.3
[16] stringr_1.3.0 yaml_2.1.18 compiler_3.4.3
[19] htmltools_0.3.6 knitr_1.20 </code></pre>
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